|No, you shouldn't run two AV's in the background together. They will fight each other and slow down your machine, possibly causing other issues too. It's OK to have any program in addition to your AV so long as you only run it on demand (when you suspect something has crept through).|
I prefer MSE and to have MalwareBytes on demand but that's just my opinion.
Please come back and let us know how you got on.